Math

QuestionAnalyze the function values for f(x)f(x): f(2)=400f(2)=400, f(4)=100f(4)=100, f(6)=200f(6)=-200, f(8)=500f(8)=-500, f(10)=800f(10)=-800. What pattern do you see?

Studdy Solution

STEP 1

Assumptions1. The given table represents a function f(x)f(x), with xx as the independent variable and f(x)f(x) as the dependent variable. . The function f(x)f(x) is a linear function, i.e., it can be represented in the form f(x)=mx+cf(x) = mx + c, where mm is the slope and cc is the y-intercept.
3. The slope mm and y-intercept cc are constants.

STEP 2

First, we need to find the slope of the function. The slope is the change in f(x)f(x) divided by the change in xx. We can calculate this using the formulam=Δf(x)Δxm = \frac{\Delta f(x)}{\Delta x}

STEP 3

Now, plug in the given values for f(x)f(x) and xx to calculate the slope. Let's use the first two points (2,400)(2,400) and (,100)(,100).
m=f(x2)f(x1)x2x1=1004002m = \frac{f(x2) - f(x1)}{x2 - x1} = \frac{100 -400}{ -2}

STEP 4

Calculate the slope.
m=3002=150m = \frac{-300}{2} = -150

STEP 5

Now that we have the slope, we can find the y-intercept cc by rearranging the formula of the linear function and plugging in the values for f(x)f(x), mm, and xx. Let's use the point (2,400)(2,400).
c=f(x)mxc = f(x) - mxc=400(150×2)c =400 - (-150 \times2)

STEP 6

Calculate the y-intercept.
c=400(300)=700c =400 - (-300) =700

STEP 7

Now that we have the slope and the y-intercept, we can write the equation of the function f(x)f(x).
f(x)=150x+700f(x) = -150x +700This is the function that fits the given data.

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